Quartic polynomials with a given discriminant

Abstract

Let $0\ne D\in  Z$ and let $Q_D$ be the set of all monic quartic polynomials $x^4 + ax^3 + bx^2 + cx + d \in  Z[x]$ with the discriminant equal to D. In this paper we will devise a method for determining the set $Q_D$. Our method is strongly related to the theory of integral points on elliptic curves. The well-known Mordell’s equation plays an important role as well in our considerations. Finally, some new conjectures will be included inspired by extensive calculations on a computer.